Question

Solve the system of equations by finding the reduced row-echelon form of the augmented matrix for the system of equations? 9x-4y+z=-4 -x+2y-3z=20 4x+4y-z=43

Answer 1

`x = 3, y=7, and z = -3`

Explanation:

The augmented matrix is:

`[ (9,-4,1,|,-4), (-1,2,-3,|,20), (4,4,-1,|,43) ]`

Multiply row 1 by 8 and add it to row 1:

`[ (1,12,-23,|,156), (-1,2,-3,|,20), (4,4,-1,|,43) ]`

Add row 1 to row 2:

`[ (1,12,-23,|,156), (0,14,-26,|,176), (4,4,-1,|,43) ]`

Multiply row 1 by -4 and add it to row 3:

`[ (1,12,-23,|,156), (0,14,-26,|,176), (0,-44,91,|,-581) ]`

Multiply row 2 by `7/2` and add it to row 3:

`[ (1,12,-23,|,156), (0,14,-26,|,176), (0,5,0,|,35) ]`

Swap row 2 and row 3:

`[ (1,12,-23,|,156), (0,5,0,|,35), (0,14,-26,|,176) ]`

Divide row 2 by 5:

`[ (1,12,-23,|,156), (0,1,0,|,7), (0,14,-26,|,176) ]`

Multiply row 2 by -14 and add to row 3:

`[ (1,12,-23,|,156), (0,1,0,|,7), (0,0,-26,|,78) ]`

Divide row 2 by -26:

`[ (1,12,-23,|,156), (0,1,0,|,7), (0,0,1,|,-3) ]`

Multiply row 3 by 23 and add to row 1:

`[ (1,12,0,|,87), (0,1,0,|,7), (0,0,1,|,-3) ]`

Multiply row 2 by -12 and add to row 1:

`[ (1,0,0,|,3), (0,1,0,|,7), (0,0,1,|,-3) ]`